Advisor:  Dr. Michael Kirby

Committee:  Dr. Henry Adams, Dr. Chris Peterson, Dr. Chuck Anderson

Title:  Schubert Variety of Best Fit with applications and Across Domains Sparse Feature Extraction

Abstract:  The conten of my PhD proposal is three-fold. In the first part, we propose a geometric framework using the Schubert variety to represent a collection of subspaces within a fixed vector space. Specifically, given a set of  $l$-dimensional subspaces$V_1, \dots, V_r$ of $\mathbb R^n$, represented as the column spaces of matrices $X_1, \dots, X_r$, our goal is to determine a representative matrix $K \in \mathbb R^{n \times k}$. This matrix $K$ is designed so that each subspace $V_i$ intersects, or nearly intersects, the span of $K$’s columns in at least $c$ dimensions. The solution of the underlying non-convex optimization problem, implemented through the introduction of an “abstract node” trained with PyTorch, yields applications in various ML tasks, including classification and quantization algorithms. Additionally, for a variety of existing ML/DL methods it enables the treatment of subspaces as vectors in Euclidean space. When compared to similar methods, our experimental results demonstrate superior performance across diverse methods and data domains. In the second part, we introduce a novel “across domains” sparse feature extraction framework. This design leverages two domain-specific VAE-based neural subnetworks with a shared sparsification layer and a common classifier subnetwork to align domains in latent space. Tested on RNA data samples from two different tissues—liver and spleen—our method successfully identifies new features that were not detectable in single-domain feature extraction experiments, while also capturing most of the domain-specific discriminative features. Finally, the third part involves my plans for the remainder of my PhD program. These plans include further experiments with our current developments and exploration of a new method approximating the geodesics on the Flag manifold.

Join Zoom Meeting:  https://zoom.us/j/93377682483?pwd=dTd0QlhHcEl3bEszZUdPUlZlcUtLQT09

Meeting ID: 933 7768 2483

Passcode: 161931